Numerical study of a family of dissipative KdV equations

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January 2013, 12(1): 519-546. doi: 10.3934/cpaa.2013.12.519

Numerical study of a family of dissipative KdV equations

Jean-Paul Chehab ^1, and Georges Sadaka ^1,

1.	LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France, France

Received April 2011 Revised December 2011 Published September 2012

Abstract
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The weak damped and forced Korteweg-de Vries (KdV) equation on the 1d Torus have been analyzed by Ghidaglia[8, 9], Goubet[10, 11], Rosa and Cabral [3] where asymptotic regularization eects have been proven and observed numerically. In this work, we consider a family of dampings that can be even weaker, particularly it can dissipate very few the high frequencies. We give numerical evidences that point out dissipation of energy, regularization eect and the presence of special solutions that characterize a non trivial dynamics (steady states, time periodic solutions).

Keywords: Korteweg-de Vries equations, long time behavior, damping.

Mathematics Subject Classification: 35B10, 35B40, 35Q53, 65M12, 65M70.

Citation: Jean-Paul Chehab, Georges Sadaka. Numerical study of a family of dissipative KdV equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 519-546. doi: 10.3934/cpaa.2013.12.519

References:

[1]	M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems, Communications on Pure and Applied Analysis, 7 (2008), 211-227.
[2]	M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation, in preparation.
[3]	M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278. doi: 10.1016/j.physd.2004.01.023.
[4]	C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes, (French) [Multilevel schemes for waves equations], ESAIM Proc., 27, EDP Sci., (2009), 180-208. doi: 10.1051/proc/2009027.
[5]	J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings, Journal of Scientific Computing, 20 (2004), 159-189.
[6]	J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations, preprint, INRIA report, HAL, inria-00529227, (2011).
[7]	D. Dutykh, Modélisation mathématique des Tsunamis, (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007.
[8]	J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eq., 74 (1988), 369-390. doi: 10.1016/0022-0396(88)90010-1.
[9]	J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359. doi: 10.1006/jdeq.1994.1071.
[10]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.
[11]	O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163.
[12]	D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications," SIAM Philadelphia, 1977. doi: 10.1115/1.3424477.
[13]	S. Mallat, "A Wavelet Tour of Signal Processing," Academic press, 1998.
[14]	A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Cambridge University Press, 2005.
[15]	L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case, preprint, arXiv:1005.4805.
[16]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping, Physics of fluids, 12 (1969), 2388-2394. doi: 10.1063/1.1692358.
[17]	E. Ott and R. N. Sudan, Damping of solitary waves, Physics of fluids, 13 (1970), 1432-1435. doi: 10.1063/1.1693097.
[18]	A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation, IMA J. Num. Anal., 20 (2000), 235-261. doi: 10.1093/imanum/20.2.235.
[19]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.
[20]	S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcialaj Ekvacioj, 54 (2011), 119-138. doi: 10.1619/fesi.54.119.
[21]	S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations, Asymptot. Anal., 68 (2010), 155-186.

show all references

References:

[1]	M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems, Communications on Pure and Applied Analysis, 7 (2008), 211-227.
[2]	M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation, in preparation.
[3]	M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278. doi: 10.1016/j.physd.2004.01.023.
[4]	C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes, (French) [Multilevel schemes for waves equations], ESAIM Proc., 27, EDP Sci., (2009), 180-208. doi: 10.1051/proc/2009027.
[5]	J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings, Journal of Scientific Computing, 20 (2004), 159-189.
[6]	J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations, preprint, INRIA report, HAL, inria-00529227, (2011).
[7]	D. Dutykh, Modélisation mathématique des Tsunamis, (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007.
[8]	J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eq., 74 (1988), 369-390. doi: 10.1016/0022-0396(88)90010-1.
[9]	J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359. doi: 10.1006/jdeq.1994.1071.
[10]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.
[11]	O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163.
[12]	D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications," SIAM Philadelphia, 1977. doi: 10.1115/1.3424477.
[13]	S. Mallat, "A Wavelet Tour of Signal Processing," Academic press, 1998.
[14]	A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Cambridge University Press, 2005.
[15]	L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case, preprint, arXiv:1005.4805.
[16]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping, Physics of fluids, 12 (1969), 2388-2394. doi: 10.1063/1.1692358.
[17]	E. Ott and R. N. Sudan, Damping of solitary waves, Physics of fluids, 13 (1970), 1432-1435. doi: 10.1063/1.1693097.
[18]	A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation, IMA J. Num. Anal., 20 (2000), 235-261. doi: 10.1093/imanum/20.2.235.
[19]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.
[20]	S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcialaj Ekvacioj, 54 (2011), 119-138. doi: 10.1619/fesi.54.119.
[21]	S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations, Asymptot. Anal., 68 (2010), 155-186.

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